One of the peculiarities of two-dimensional (2D) quantum systems is the existence of particles which are
neither bosons nor fermions, but whose exchange statistics interpolates between the two.
Since these particles can have any statistical phase, they were termed anyons [1]}.
Anyons are a direct consequence of the fact that the process of exchanging two identical particles is not merely a permutation of their quantum numbers, but an adiabatic interchange of the positions of the particles.
The dimension of a quantum system determines the possible
indistinguishable particles it can support [2]}, [3]}.
The topology of 2D systems, unlike the more familiar 3D systems, makes the interchange
of two indistinguishable quantum particles nontrivial, leading to the very existence of anyons.
For two identical noninteracting anyons whose energy spectrum is nondegenerate, the exchange
phase is a special case of the geometrical (Berry) phase [4]}.
In this case, the Berry phase is not only geometrical, but is also topological in nature. These type of anyons are called Abelian anyons.
On the other hand, a prerequisite for non-Abelian anyons is a degenerate ground state manifold [5]},
such that braiding of anyons corresponds to a unitary transformation (rotation) within the subspace manifold.
This rotation is described by a unitary Wilczek-Zee matrix [6]}, which is a direct generalization of the Berry phase.
